Divisibility rules

Learning Objective:

• we are learning to find out if a particular whole number is a multiple of another smaller number.
• success criteria: by the end of this lesson you should be able to quickly work out whether 8971957 is in the 9 times-table.

Investing:

• This is useful because it is a skill that allows us to find factors of a number and write a number in its prime factorisation.
• A functional (real-life) application is the everyday act of rounding amounts of money to the nearest £10 or $5.
• A less obvious use of understanding divisibility is parity bits and error detection and correction. This is where the divisibility (and similar) properties of a binary number transmitted as an electrical signal are used to check that a signal has been correctly received. Bit parity is used in coding music onto audio compact discs using and in transferring data over computer networks.
• Other organisations use the ideas of divisibility and check digits to generate bank account numbers, UPC barcodes for goods and ISBN codes for books. Because these organisations make sure that the numbers they use have certain properties, such as having a consistent remainder when divided by 11, an understanding of the divisibility rule for multiples of 11 can allow users to recover information about numbers that have been incorrectly written or mis-read. (A nice AMS mathematical moments poster on ISBN)

• Knowing divisibility rules leads to a better understanding of multiplication and division, prime numbers, factorising algebra, simplifying fractions, simplifying ratios and lots of other arithmetic and algebra skills. Knowing some of the rules may make it easier to learn some of the mulitiplication tables too.
• A 'free gift' with this skill is that we will be able to use a 'base nine checking trick' to check our answers to long addition sums and hard multiplication problems.
• An understanding of divisibility and error checking could be useful for careers in banking, network engineering, computing, audio and media engineering, telecommunications, data security.

Preparing:

• Can you add up single digit numbers? Then you're ready!

Divisibility by 2:

• Can you figure it out yourself from these examples?

• Us this random number generator to try out a few:

Modeling:

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:
• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...